Slides_2010_03_08

# Slides_2010_03_08 - Applied linear algebra and numerical...

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Unformatted text preview: Applied linear algebra and numerical analysis Session 26 Prof. Ulrich Hetmaniuk Department of Applied Mathematics March 8, 2010 Eigenvalue problems • Eigenvalue problems give insight into the behavior of evolving systems governed by linear equation. • Example: Study of resonance • Swing • Musical instruments (drum, Helmholtz resonator) • Bridges • Many other applications... PageRank • Algorithm used by Google to rank web pages. • The PageRank values are the entries of the dominant eigenvector of the modified adjacency matrix. Review of Linear Dependence Are the following set of vectors linearly dependent? • 1 1 , 3 2 • 1 , 3 • 1 , Review of Linear Dependence Consider the matrix A = 3 0 0 2 . Are the following set of vectors linearly dependent? • x = 1 1 , Ax = 3 2 • x = 1 , Ax = 3 • x = 1 , Ax = 2 • x = x 1 x 2 , Ax = 3 x 1 2 x 2 with x 1 6 = , x 2 6 = • = , A0 = Linear Dependence – Eigenpairs Consider a non-zero matrix A ∈ R n × n ....
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Slides_2010_03_08 - Applied linear algebra and numerical...

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